/sci/ - Science & Math

You can map any line segment to the entirety of any n-dimensional space.

>>4390650
sorry, meant infinitesimally large shape. I forgot an adjective there.

>>4390671
Proof of this? I was led to believe that if you move up a dimensional from <span class="math">\mathbb{R}^n \to\mathbb{R}^n^+^1[/spoiler], the set of all numbers becomes infinitesimally larger.

A two dimentional fractal will have a finite area, but infinite _perimeter_.

>>4390686
<span class="math">\mathbb{R}^n \to\mathbb{R}^{n+1}[/spoiler]

>>4390689
okay, then let's use gabriel's horn as an example

I'm confused by your use of the word "infinitesimal".

Do you mean "infinite"?

>>4390706
yeah, sure. just has to be huge.

Basically can I take the area of a two-dimensional figure and map it one-to-one with the volume of a three-dimensional figure?

>>4390686

http://en.wikipedia.org/wiki/Space-filling_curve

Remember that all rationals can be expressed as the ratio between two integers, but the cardinality of the rationals and integers is the same.

You can't map an n-dimensional real object to a (n+1)-dimensional. You're probably thinking of the fact that you can map the natural numbers to any vector space over the natural numbers, but that does not work for reals. If you don't believe me, try describing the position inside a two-dimensial plane, with one number. In fact the dimension of an object is informally defined as as the number of co-ordinates needed to describe a point in that object.

>>4390721

http://en.wikipedia.org/wiki/Space-filling_curve#Proof_that_a_square_and_its_side_contain_the_same_n
umber_of_points

I'll give you the honor of editing this, since you know it's incorrect.

>>4390778
Well I'll be a hog-monkey's uncle.
It turns out everything I thought I knew about mathematics is wrong. I guess I'll have to drop out of school and travel the rails with my hobo bindle.

>>4390778
>manifold

not the volume